S-i-1-i-n-n-R-Can-S-be-solved- Tinku Tara June 3, 2023 Arithmetic 0 Comments FacebookTweetPin Question Number 4668 by FilupSmith last updated on 20/Feb/16 S=∑∞i=1in,n∈RCanSbesolved? Answered by Yozzii last updated on 20/Feb/16 S(N)=∑Ni=1n0.5i=n∑Ni=1(n0.5)i−1S(N)=n((n)N−1)n−1S(N)=(n)N+1−nn−1(1)Forn⩾1⇒n⩾1⇒(n)N+1→∞asN→∞.limN→∞S(N)=(n)∞−nn−1=∞.S(∞)doesnotexist.(2)If0⩽n<1⇒0⩽n<1⇒(n)N+1→0asN→∞.∴limN→∞S(N)=(n)∞−nn−1=n1−nS(∞)exists.(3)For−1<n<0letn=−a,0<a<1.⇒n=−a=ia=a(cos0.5π+isin0.5)=e0.5iπa⇒(n)N+1=e0.5iπ(N+1)(a)N+1Since0<a<1⇒(a)N+1→0asN→∞.∴limN→∞S(N)=0×e∞i−nn−1=n1−nS(∞)=ia1−ia=ia(1+ia)1+aS(∞)=−a+ia1+a=n+i−n1−n.⇒S(∞)∈C.(4)Forn⩽−1⇒(n)N+1→ei∞ifn=−1.⇒a=1.S(N)=(n)N+1−nn−1=(ia)N+1−iaia−1=ia((ia)N−1)ia−1=ia(1−ia)(1−(ia)N)1+a=(a+ia)(1−(ia)N)1+a=a−a(ia)N+ia−(ia)N+11+a=a+ia1+a−(ia)N(a+ia)1+aS(N)=a+ia1+a(1−a0.5Ne0.5πiN)limN→∞S(N)=1+i2(1−ei∞)whichisundefinedsinceei∞isindeterminate.(5)Ifa>1thenlimN→∞S(N)=a+ia1+a(1−∞ei∞)∞ei∞isundefined⇒S(∞)isundefinedforn⩽−1. Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Question-70198Next Next post: Question-135742 Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.